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%I #27 Feb 28 2022 03:06:02
%S 1,1,1,2,4,12,38,150,648,3218,17476,105150,686012,4858602,36966912,
%T 301473146,2619548392,24181633098,236214717380,2435055354798,
%U 26413257186156,300761256373298,3586799346195376,44710179730250874,581430401788576664,7874808892125746226
%N E.g.f. A(x) satisfies: 1-x = Sum_{n>=0} Laplace(A(x)^n) * (-x)^n.
%C Here Laplace( Sum_{n>=0} c(n)*x^n/n! ) = Sum_{n>=0} c(n)*x^n.
%H Vaclav Kotesovec, <a href="/A213058/b213058.txt">Table of n, a(n) for n = 0..220</a> (terms 0..100 from Paul D. Hanna)
%e E.g.f.: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 4*x^4/4! + 12*x^5/5! + 38*x^6/6! +...
%e The table of coefficients in the Laplace transform of A(x)^n begins:
%e n=0: [1, 0, 0, 0, 0, 0, 0, 0, 0, ...];
%e n=1: [1, 1, 1, 2, 4, 12, 38, 150, 648, ...];
%e n=2: [1, 2, 4, 10, 30, 104, 420, 1896, 9632, ...];
%e n=3: [1, 3, 9, 30, 114, 486, 2316, 12210, 70632, ...];
%e n=4: [1, 4, 16, 68, 316, 1608, 8936, 54024, 353496, ...];
%e n=5: [1, 5, 25, 130, 720, 4280, 27330, 187230, 1372640, ...];
%e n=6: [1, 6, 36, 222, 1434, 9792, 70908, 544800, 4437288, ...];
%e n=7: [1, 7, 49, 350, 2590, 20034, 162680, 1389066, 12474672, ...];
%e n=8: [1, 8, 64, 520, 4344, 37616, 339216, 3193200, 31407632, ...];
%e n=9: [1, 9, 81, 738, 6876, 65988, 655326, 6752934, 72308376, ...]; ...
%e such that the alternating sums of antidiagonals vanish for n>1:
%e 1 - 1 = 0;
%e 1 - 2 + 1 = 0;
%e 2 - 4 + 3 - 1 = 0;
%e 4 - 10 + 9 - 4 + 1 = 0;
%e 12 - 30 + 30 - 16 + 5 - 1 = 0;
%e 38 - 104 + 114 - 68 + 25 - 6 + 1 = 0; ...
%o (PARI) {a(n)=local(A=[1,1]);for(i=1,n,A=concat(A,0);A[#A]=Vec(sum(m=1,#A,serlaplace(serconvol(Ser(A),exp(x+O(x^#A)))^m)*(-x)^m))[#A]);A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A204190.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jun 06 2012
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